what are polynomials and their identity plzz help
Answers
Answer:
Polynomial Identities : An algebraic expression in which the variables involved have only non negative integral powers is called polynomial.
For factorization or for the expansion of polynomial we use the following identities.
Important Polynomial Identities :
1) ( x + y )2 = x2 + 2xy +y2
2) ( x – y) 2 = x2 – 2xy + y2
3) (x + y)(x – y) = x2 – y2
4) (x + a)(x + b) = x2 +(a + b)x + ab
5) (x + y) 3 = x3 + 3x2y + 3xy2 + y3 = x3 + y3 +3xy(x +y)
6) (x - y) 3 = x3 - 3x2y + 3xy2 - y3 = x3+ y3 -3xy(x –y)
7) (x + y + z) 2 = x2 + y2 + z2 + 2xy + 2yz + 2zx
8) x3 + y3 = (x + y)(x2 – xy + y2)
9) x3 - y3 = (x - y)(x2 + xy + y2)
10) x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz – zx)
If x + y + z = 0 , then x3 + y3 + z3 = 3xyz
Some solved examples :
Expand the following using Polynomial Identities.
1) (2a + 5) 2
Solution : (2x + 5) 2 = (2a) 2 + 2(2a)(5) + 5 2
[ using the identity ( x + y ) 2 = x 2 + 2xy +y 2 ]
= 4a 2 + 20 a + 25
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2) ( b + 6)(b - 6)
[ using the identity (x + y)(x – y) = x 2 – y 2 ]
( b + 6)(b - 6) = b 2 - 6 2
= b 2 - 36
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3) ( 3a - 4) 3
[using the identity(x - y) 3 = x 3 - 3x 2 y + 3xy 2 - y 3 ]
( 3a - 4) 3 = (3a) 3 - (3a) 2 (4) + 3(3a)(4) 2 - 4 3
= 27a 3 -72a 2 + 36a - 64
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Factorize the following using Polynomial Identities :
1) Factorize: 64a 3 - 27b 3 - 144a 2 b + 108ab 2 .
Solution :
64a 3 - 27b 3 - 144a 2 b + 108ab 2
= (4a) 3 - (3b) 3 - 36ab(4a- 3b)
= (4a) 3 - (3b) 3 -3(4a)(3b)(4a -3b)
= (4a - 3b) 3 [ using x 3 + y 3 -3xy(x –y) ]
= (4a -3b)(4a -3b)(4a -3b)
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2)Evaluate : (104) 3
(104) 3 = ( 100 + 4) 3
= = (100) 3 + (4) 3 + 3(100)(4)(100 + 4) [Using Identity V]
= 1000000 + 64 + 124800
= 1124864
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3) Evaluate : (–12) 3 + (7) 3 + (5) 3
Solution : (–12) 3 + (7) 3 + (5) 3
From the above we can see that -12 + 7 + 5 = 0
(–12) 3 + (7) 3 + (5) 3 = 3(-12)(7)(5) [Using identity 10]
= -1260
Answer:
polynomial identities r equations that r true for all possible values of variable.....
Step-by-step explanation:
a polynomial is a algebraic expressions with a finite number of terms . these terms are in the form of "AXN" where a is a real number "x" means to multiply and "n" is a non negative integer